[[Ring theory MOC]]
# Dedekind domain

A **Dedekind domain** $R$ is an [[integral domain]] such that #m/def/ring 

1. $R$ is [[Noetherian ring|Noetherian]];
2. $R$ is [[Integrally closed domain|integrally closed]];
3. $R$ has [[Krull dimension]] $1$, i.e. every nontrivial [[prime ideal]] is [[Maximal ideal|maximal]], and there exists a nontrivial prime ideal.

## Results
 
- [[A Dedekind domain admits UFI]], moreover it is a [[Unique factorization domain|UFD]] iff it is a [[Principal ideal domain|PID]].
- [[Fractional ideals of a Dedekind domain form an abelian group]]
- [[Ring of integers of a number field]] form a Dedekind domain and lattice.
- [[Ideals of a Dedekind domain need at most two generators]]
- [[A Dedekind domain with finitely many prime ideals is a UFD]]
- [[A Dedekind domain is a UFD iff its ideal class group is trivial]]
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